Barrycentric coordinates for a polytope

salman7866

For the past few weeks, I have been searching about this topic: Suppose we are given a convex polytope having vertices say \begin{equation} A_1, A_2,...,A_n \end{equation} where each \begin{equation}A_i, i=1,...,n \end{equation} represent a matrix - Infact its a convex polytope whose vertices are matrices. How can we verify if a given a matrix say \begin{equation} A_t \end{equation} can be written as a convex combination of the vertices.

To elaborate more: I am constructing this polytope to encompass a time-varying matrix; say denoted by A(t) where the variable "t" is varying between some upper and lower bounds t_{min} and t_{max}. Using these bounds of t, I find vertices for a polytope denoted by \begin{equation}A_1,...,A_n \end{equation}. Now how do I prove that given A(t) where \begin{equation}t_min \leq t \leq t_max \end{equation} can be written as convex combination of the vertices \begin{equation}A_1,...,A_n \end{equation}. Any help would be appreciated.